X-Rays diffraction study of the plastic zone

Enginkng

Fmcrwe

Mccktuks.

1976, Vol. 8, pp. 691-700.

Pespn?a

Pas.

P&&d

in Great Britaia

X-RAYS DIFF~CTION STUDY OF THE PLASTIC ZONE HENRIJEAN LATIERE DirecteurdeRechercheTiNaire,CentreNati~delaRechercheScigltifique,LaboratoiredeM~queet d’Acoustique, 31,cheminJoaepbAiguier,13274Marseillecf?dex2,France At&net-The energyabsorbedby the plasticzone nearthe tip of fatiguecracksis measuredby X-rays diffraction methodinthecaseof AU2GNandthe “butterfly wings” axesinthecaseof AGS,areplotted.

INTRODUCTION ALLTHEmechanical structures contain cracks. The most important industrial problem is to know

the conditions and above all the speed of the development of these cracks. This speed varies with, among other things, the strain energy adsorbed at the tip of the crack. The more this strain absorbs energy the more slowly the crack grows. The first problem is, thus, to find this energy and the second is to determine the thickness and the orientations of the plastic zone[ll. As early as the first readingson these problems, what strikes us most, is the constant presence of the Williams-Irwinequations which show that the strain is first proportions to the reverse square root of the distance to the tip of the crack, r-I’*, and then to a certain R factor. By considering the moment of these theoretical results, any research worker, even ex~~en~st and perhaps speciallyin this case, must clearly understand how these results are obtained. So one writes in detail, completely but as easily as possible, the chain of reasoning that leads to W.I. equations (see Appendix).It is noteworthy that from the first elements of the theory of elasticity, one can reach simple expressions physically verifiable.These verificationsare the scope of this paper: to measure the energy absorbed by and the “radius” of, the plastic zone into AU2GN,and to plot the axes of “butterfly wings” into AGS. 1.ENERGY AWORRRDBY THEPLASTIC ZONE ATTHE TIP OF A FATIGUE CRACK IN AU2GN 1. heretics

~pression of the sfrain energy The principal hypothesis of Dugdale[2] is that the material in the yielded zone is under a uniform tensile yield stress Y. With this postulate, Goodier and Field [3]arrive to the expression for a unit thickness:

where, for AU’LGN:K = (3- u)/(l + V)plane stress case, v = Poisson’s ratio = 0.3, E = Young modulus = 7200Hb = 72 * lo” Pascal, Y = Yield stress = 38Hb = 38 * 10’Pascal, f(,, = (t = T/Y, T, normal tensile stress) = 0.110, value given by the authors, W, = absorbed plastic energy, I = half length of the crack. Finally, in the case of AU2GN plates the theory gives: 1%1.37 7 10" 4**ttv *I 0110 S-r 7r **' (7.10"> ' = 0.26 *10’I barye (cgs) or 0.26 -10’i Pascal @.I.) 1.2. Experimental expression (a) Equi-energycurves.X-raysstudy. Wood[4, S],McClintockandIrwin[6],andShall[7]made use of X-raysd&action to study fatiguedamage,Felixand Geiger[8],Debye-Scherrerdiagramsto studyb~~e~p~eof steel~d,sotodisti~shNpt~es bydeformationof discohesive0nes.T~ 691

H. J. LATIERE

692

et al. [9], Yokobori et al. [ lo], used Hirsch technique (microbeam)and brought out sub-structure state.HellwigandMacherau11l] studythe stressesdistributionconveyedby Debye-Scherrerradius increase. One utilizesfirst Laue method with a tungsten target under a tension of 60 KV. to have a good X background. The direct diagram, with a 4jlOmm diameter collimator, distance sample-film 40 mm gives a concen~ation of Laue spots wrapped up into the region of the m~imum X-rays bac~ound we see the gatheringof these spots (lowerFig. 1),when the grain of the metal is good, for instance far for the crack. If, on the contrary, the grain is strained, each of its reticular planes turned up under Braggangleof incidence,gives not a ponctual point but, on account of bendingan asterism trail (upper part Fig. 1).These two pictures correspond, the upper part to open sheet, the lower part to the edge of the crack. They are two extreme cases but we can distinguishbetween intermediate points (Fig. 2), even if the difference is weak, (Fig. 3). So, to trace equi-straincurves (Fig. 4) around the crack, it is sufficientto take diagramsin the vicinity of the crack and to compare them with one another. It is also possible to determinate the absorbed energy, ascribed to the unit surface of the sheet, in each of its points. It is sufficientfor that to compare each diagram with standard diagrams. To obtain a standard diagram one proceeds in this way: a non-notched sample is submitted to a weak pull then released. The curve so obtained gives the sample absorbed energy. This energy is divided by the sample strained surface to obtain the absorbed energy per unit surface of the sample (not fracture surface). The X-ray diagram of this sample is the standard diagram. We repeat the process as many times as required by the needed accuracy with higher and higher loads. A running of standard diagramsis so obtained. Then, each X diagram of the plastic zone can be compared with this running to determine the energy absorbed by unit surface in the correspondingpoint of this plastic zone. A simple summation gives the whole absorbed energy. For a T6 state AU2GN sample, 2 * 10-lm 13-23Hb, I find 4 J for the quarter of the crack. width, 16. lo-‘m thick, frequence 60, 0;11px: We see, (Fig.4) that the equi-energizingcurves are straight lines. That is in agreementwith the theory. As a matter of fact (Appendix)

(b) Setting up of the experimental expression according to the experimental curves. After these curves, the fourth part of the absorbed energy, when the crack is 21long, is proportional to I *. In Plastic zone

AU2GN

Fig. 4. E&energy curves around the crack. EXtellSh

% g.2 00.4 A 0.6 no.9 A 3.5 04.2 CM A If OWure~

Absorbedenergy (XIQ’ Joule) bymm*of surface sample 24 0 158 318 884 1657 2037 2txm(?)

UK)(?)

Fig.l. (Left)LaueSpote,whcatheBrsinispood,openshect;(Right)Asterismtroils,wheathc~isstrained, edge of the crack.

Fig. 2. Asterism trails are longer when the grain is more strained (right). ‘fix left figure is given by a point of the sheet at 2mm of tbe edge crack, the right at 0.5mm.

Fig. 3. The points of the sheet are at 4 (left) and 3 mm (right) of the edge of the crack.

Fii. 5. Back Debybscherrer X diagrams of the crack edge (left) and of the plane sheet (right). [Facing page 6921

Fig. 6. The Debye-!%herrer circle is more contiuuouswbeu the point consideredis nearertbe edge of the era& 0.5 mm for the left diagram,1 mm for the right.

Fii. 7. Back Laue X diagramstaken on a perpendiculartine to the fracture,at 0,0.5,1,2,3 amI6 mm of the crackedge.

lofthe

Fig. 9.

b. 10. Determination of the strainaxes neara OOtch.

X-raysdiffractionstudyof the plasticzone

693

respects, in fist approach, for thin plates, for which we stay in plane stresses domain, this energy is proportional to the thickness e of the plate. Let WIbe the absorbed energy, A a constant to determinate, we can write:

other

tW,= Ae12. To dete~nate A we consider the case of the broken sample: e = 16- lo-* m, t = IO-’ m Ww1 = 4 J, so 4=A * 16. IO-‘*IO-’ or A=? and #+T.

16. *O”Iz

tw;=4* lo’i’. For the balf crack extension dl $dWg =8.lo2idf. The energy absorbed by surface unity of the crack is:

or $ d$ for unit thickness = 5 - 107 cgs .

The experimental results are bigger than theoretical ones by a factor 2. To obtain theoretical expression, the hypothesis is that into the yielded zone there is a uniform tensile stress that is the yield stress Y. But, the experimental works show that the stress varies with the strain. And these variations are properly a method to measure the strain by the stress (Hellwigand Macherau[ 111). And it is well known that the strain into the plastic zone increases with the reverse distance to the edge crack. So, it is reasonable to think that the stress is higher than Y of the plate in the very rear of the crack, because the yield stress increases with strain hardening. And the theoretical calculate gives only a lower limit. 2. RADIUSOF THE PLASTIC ZONE The predicted cyclic plastic zone within monotonic plastic zone, by Rice[121was revealed by

etching technique, Hahn1131and microhardness measurements, Bathias and Pelloux[l4]. The X-ray method here used is nearly in accordance with this result (Fig. 4). It is sutllcientto control this to make a section perpendicular to x-axis of the curves. To increase the accuracy we are building an adjustable microcollimator. The irradiated area on the specimen will be, as it is observed at the first testings, 30 p. Moreover it is easier than classicalmicrocollimatorto adjust. We think to be soon able to measure plastic zone near the begimring of the crack. That is important because the theory (see Appendix) is accurate near the origin of the crack. But I would like to point out a method to measure the r of the end of the appendix. It is sutllcientfor this to take back Debye-Scherrer X diagrams,in the same cautions as that of the direct ones. We obtain circles coKes~~g to the W c~~te~stic emission. The D.S. circles are ~n~uous when they correspond to the crack edge, very strain hardened (lower Fig. 5) and dotted when they correspond to the plane sheet (upper Fig. 5). It is possible, as with direct diagrams, to increase the accuracy, to distinguish between two points of the plastic zone, the

6%

H. J. LATIERE

most strain hardened ones (Fii. 6). The lower diagramof this @ure correspond to a point of the pk&ic zone more strained than that of the upper ones. The point that corresponds to the radius t referred to at the end of the Appendix,gives a diagram located between the two diagramsin the middle of Fig. 7. The six diagramsof this figure are taken on a perpendicularline to the fracture. The distances of the ~ffer~nts points to the edge of the crack are 0; 0.5; 1; 2; 3 and 6 mm, At 1 mm the circle is still continuous. And at 2 mm it is dotted. The limit can be located at 1.5mm. This limit corresponds to the theoretical radius.

3, OBLATIONS OF THE PLASTIC ZONE“COMPREHENSIVEI3UGRAM” A finer method to determine the strain in the plastic zone Iies in making this study on coarse grains samples. Figure 8 shows the p~n~iple of the method*A non-strained grain, entirely lying tmder a ~~y~~o~~~ X-raysbeam gives Laue spots the shapes of which are ho~~et~ to that

Fi. 8. PrimSpIeof the ~~s~e

disjpm

of the corresponding grains. Then a deformation in a region of the grain appears in the horno~e~c region of the spot. Figure 9 shows the diagramsof three grains. On the left the grains are in the originatestate. On the right they have been submitted to lo’ cycles with Q- = 30MPa. It is easy to see the strong locaiised perturbation of the spot which corresponds to the strain of the grain in a similar position. Fiie IOis an applicationof this method to determine the direction of the axes of the plastic zone. In the middle of this figurethere is the sketch of the sample surface under the X-rays.We see the two notches. The horizontal lines are the boundaries of the grains that are in the direction of the length of the sample. Each X-rays spot is identifiedby the same letter which identifiesthe grain. We see, in this Fig* IO,that the grain “d” for instance is placed at the end of the upper notch. At the beginningof the fatigue, the deformation is strong in this region. So, the spot given by this grain is disturbed in his middle.This spot “‘d”is under the sketch or at the North-Westof the Fig. 10.On the contrary, the spot e, at the North-East mainly, is not disturbed in his middle, but in two points that divide the spot into three parts. So, it is possible to lay out the directions of the maximum deformation of the sampIe at the end of a notch.

X-rays diffraction study of the plastic zone

695

4.CONCLUSION At first in this paper, one proposes three X-raysmethods to determine respectively, the strain energy, the radius, and the directions of the plastic zone. These experimental results are in good agreement with the theory. Secondly, one tries to give, as simple as possible a survey to understand W-I equations. What is, for an ex~~ment~ist, the origin of these equations? what is the chain of reasoning to reach them ~Ap~nd~). Acknowtedgemenfs-The author has been helped in his work by discussions with Prof. Sertour and Prof. Bathias. He also wishes to thank hf. H. Caumon and M. D. Vovan for technical aid.

REFERENCES [l] G. Satour

and C. Bathias, La zone plastit& &fond de fissure de fatigue. ler Con8r&sFran~&~de Mtique,

17-20 Septembre 1973.A6rospatiale (France). [2] S. Dugdale, Yielding of steel sheets contain@ slits. J. Me& Phys. Solids 8, 100-104(l%O). [3] N. Goodier and F. A. Field, Plasrie Eneqy lXssipation in Crack Propagation Fracture of Solids, AS AZME, pp. 103-118,Gordon and Breach, Paris (1%3). 141W. A. Woad,Pmt.Roy. Sot. Lmifon A174,310 (1940). [5] W. A. Wood and R. B. Davies, Proc. Roy. SM. lmdon Am, 255 (1953). [6] P. A, McClintock and 0. R. Irwin, Plasticity aspects of fracture mecanics, fracture toughness testing and its applications. Am S’f’P 381, 84-113 (1%4). [7j A. Scball, Z; Me&&de, Jg, S.417, 40 (1949). [SJ W. Felix and Th. Ge&r, Sur la ~pture dcs aciers par fm@it6. Reuue Tecknkpe St&er 1, M-27 (19%). [9] S. Taim and K. Honda, X-my invest&a&m on the fatigue damage of metal. lap. Sot. Me&. Engng 4,14,230-237(1%1). [lo] T. Yokobori, K. Sato and Y. Yamquchi, X-my microbeam studies on plastic zone at the tip of the fati8ue crack. Rep. Res. Inst. Stre@h & Fmcture of Material, Tohoku Univ., Vol. 6, pp. 49-67 (1970). [ll] G. Hellwig and E. Machemucb, Die Spannun8sverteilung nake der Risspitze an8erissener Zusproben aus Ck 45. Sonderdruck aus Z&s&rift Metallkunde Band 65, Heft 1, S.75-79, (1974). [12] J. R. Rice, Mechanics of crack tip deformation and extension by fatigue. ASTM STP 415,247-311 (1%7). [13] 0. T. Hahn, R. G. Hoa8land and A. R. Rosenfleld, L.ocal yieklin8 attendin8 fat&ue crack growth. M&f/. Trans. 3, 1189-1202(1972). [14} C. Bathias and R. M. Pelloux, Fatigue.crack propagation in ma&n&c and austenitic steels. MetaL Trans. 4.1265-1273 (1973).

APPENDIX Elementary heoreticd sumey Generally, in the very manemus papers on the subject, we tind the Williams-Imin (W.I.) equations. But it is very difficult to 8et the complete explanations on how to tend these equations. My aim is to &e to en8ineers interested in this very important practical problem but who are not speck&s, the.possibility of learning very quickly essential points of the theory witbout makin a fastidious bibliiographicalwork. So, let us consider a small element of a thin plate to be in plane stresses conditions (Fig. 11). In the.case of the unit thickness plate, the sum of the X components forces act& on the element is in equilibrium state:

($+$++xdy =0 but dx dy is not null, hence we obtain (1) we obtain on the y axis: (2)

indices the 8rst points out the plane on which it acts and the second the dire&m. For instance a, acts on the plane that is perpendicularto x axis, T., acts on the plane paper&&r to x axis in the y dire&on.

H. J. LATIERE

r,+%r,t8xdx

t -u,+%u,/0xdx

I

Fig. 11. Stnasesonanelementaryblocinathinplate. we

writetbatthe sumof the moment8about 0 is nulk ~~~dy)(y~~)-(~=f~d*)dy(y+~)-(ady~ t(~,t~dX)dl(~t~)-~,~(~+~)

+(~~+~dy)~(~+~)+(~~=~)y-(~~=+~dy)~(y+dy)=O. onecimseethat: (ox-uz)y dy=O (0; _**)TcCJ (GY- T.~)xdy = 0 (a, -*,)d.x$=O fu, -0,)x dx =o -7=)y dx=o. @7X And if we neglect the terms of the third order: Y ax= dx dy? (!!$_“) (F-2)dxdyd.x $-dydx? it finally leaves

X-rays d&action study of the plastic zone

J

G,

= 7?.*

1

697

Of

The shearingstressescomponents are symmetric. Let us now consider a function f<., of the complex variable z. We can write: fCzj= PC.,, + iQ.,)

(4)

It is supposed that P ad Q have partial continuous derivatives. The question is to definite the condition for the ratio f CZ+AZ, -fw

br

(51

to tendto a Iimit whetever may be tbe way z + AZtends to z. Suppose first that the point z + AZtends to z on a line pamlIe to thcreataxis.fnthiscaseAy=Oandtberatio(5)is: ~(x+hx,y)-P(x,~)+~QOx+hr,~)-Q(x,y) Ax Ax When AZ,that is to say Ax, tends to zero, aP .aQ an+%%

P(x,Y +AY~-P~~,Y)~~Q(x,Y +b~f-Q(s~) iAy iAy and

itsIimIt value is %2+X ay i ay

if we write that the two limits are equal, we obtain:

s give by addition that

a?

aV

,x,+&F=0 and

a'Q a'Q

3+2=0

ay

’ ?P(!at?@m anda, weobtain

So, the compatibility equation becomes

(8) For a given problem, the function Q, must satisfy eqn (8) and the boundary conditions.

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H. J. LATIERE

It is obvious that Q so chosen satisfies equilibrium equations aa,

a’@

;i;;=iijG auI_ ay

a’@

aPay

a’@

ar

22=_

ax

ax2ay

a7

a%

)n=__

ay

axaf

and if we replace these values into (1) and (2) we obtain for the first members:

which both vanish. Westergaard de6ne an airy function 0 such that @, = Ret + yl,i?,

(9)

where z = x + iy is the. complex variable of the function &, and his derivatives

After the Cauchy-Rieman conditions we can write for instance aRi7

aI2

u

-=-=

ax

ay (10)

!p?pJ. -

i

After (9) and (10) we have

(11) a@,

aR&

aI,Z,

-=-+yax ax ax

(12)

(13) After (12 we can write

=+r,z, aXay=aY+y ay a%

aR&,

= - r,,,Z, + yReZ, + LZ, (14)

X-raysd&action study of the plastic zone

699

Finally, from 0, (ill, (13)and (14)we have

After Westergaard,“in a restrictedbut importantgroupof cases the normalstressesand the shearingstressin the direction of x and y can be statedin the form(15)“.Hereis the problemof the experimentalist.The choiceof Z, can givean approach of the solutionof the true phenomemm.In the case of a crack alongx-axis it is necessarythat cryand rXrbe IIUUalongthis portion of axis where lays the crack. So, if we choose

fory=Oandfor-a~x
solvesthe probiemof a stress-freecrack aiong x axis.

Let us consider: so we have

u(b‘+a)

zl=

g’n(g+ 2uyn and

for g+O. ‘l’batis to say, near the crack tip

Generally one writes

when (r is normal to the crack. And ICI z=@qid z;

=

-m

az,=-E& (-#)g-‘n ag

(21)

We put in poliv coordinates g=rP g-v2= r-wze-3m* =r

-312

thence

zz’_’

(

,q_,q

KS ,-wz+l_B

= ’ 2(2a)‘” and

canbevnitten

2

H. J. LATIERE

700

I

Y&Z;=-&sinicosicosy

I

With (16)48), (15) becomes:

1

K

a=,, = (24 a,=,,

K,

(24 K Gr =112 (274

sin-Bcos-Bcosl-e 2 2 2

(19)

K, = /iii (2?r~3”‘Z, The Wiis-Irwin equations. In plane stresses and mode J, G, and a, are respectively null and of no use. For B =0 we have:

and the radius of tbe plastic zone is obtained, if we consider that the plastic deformation takes place when u, attains the value of the elastic limit a: K,

4e. =(2m)"* or

)

r=+-(e)‘.

(